The free Banach lattices generated by ${\ell }_p$ and $c_0$

Abstract

We prove that, when $2<p<\infty$, in the free Banach lattice generated by ${\ell }_p$ (respectively by $c_0$), the absolute values of the canonical basis form an ${\ell }_r$-sequence, where $\frac{1}{r}=\frac{1}{2}+\frac{1}{p}$ (respectively an ${\ell }_2$-sequence). In particular, in any Banach lattice, the absolute values of any ${\ell }_p$ sequence always have an upper ${\ell }_r$-estimate. Quite surprisingly, this implies that the free Banach lattices generated by the nonseparable ${\ell }_p(\Gamma )$ for $2<p<\infty$, as well as $c_0(\Gamma )$, are weakly compactly generated whereas this is not the case for $1\le p\le 2$.

arXiv comment10 pages